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UPA LAW YY 


University of the State of New York Bulletin 


Entered as second-class matter August 2, 1913, at the Post Office at Albany, 
N. Y., under the act of August 24, 1912. Acceptance for mailing at 
special rate of postage provided for in section 1103, act of 
October 3, 1917, authorized July 19, 1918 


Published Fortnightly 











No. 865 ALBANY, Ney. December 1, 1926 


- a THE LIBRARY {iF THE een 


C | | 
N48uT MAY 27 1999 
no. 65 UNIVERSITY o¢ ,,, 
tL 
ARITHMETIC IN THE RURAL AND VIQ/BAGE 
SCHOOLS OF NEW YORK STATE 


LBY 


JACoB 5S. ORLEANS 


Research Associate, Educational Measurements Bureau 
AND 


EF, Eucene SEYMOUR 


Supervisor of Mathematics . 


ALBANY 
THE UNIVERSITY OF THE STATE OF NEW YORK PRESS 


1926 


G197r-N26-2000(5183)* 


THE UNIVERSITY OF THE STATE OF NEW YORK 


Regents of the University 
With years when terms expire 


1934 CuHeEsTER S. Lorp M.A., LL.D., Chancellor - - Brooklyn 
1936 ADELBERT Moot LL.D., Vice Chancellor - - - Buffalo 
1927 ALBERT VANDER VEER M.D., M.A., Ph.D.,. LL.D. Albany 
1937 CHarLtes B. ALEXANDER M.A., LL.B., LL.DI 


Litt. D. - -3- - - - - - - =~ <]9SS Tee 

1928 WALTER Guest Ketitoce B.A., LL.D.- - - - Ogdensburg 
1932 James Byrne B.A., LL.B., LL.D. - - - - - New York 
1931 Tuomas J. Mancan M.A., LL.D. - - - - - Binghamton 
1933 Witt1aAm J. Wattin M.A. - - - - - - - Yonkers 
1935 Witit1am Bonpy M.A., LL.B., Ph.D., D.C.L. - New York 
1930 WittiAM P. BAKER B.L., Litt. D. - - - - - Syracuse 
1929 Ropert W. Hicpir M.A. - - - - - - - ~- Rochester 
1938 RoLanp B. Woopwarp B.A.- - - - - - = Jamaica 


President of the University and Commissioner of Education 


Franxk P. Graves Ph:D.,: Litt. D,, Eee 


Deputy Commissioner 


Avucustus S. Downinc M.A, Pd_D., CAD Siie 


Counsel 


ERNEST EV Gorn icc 


Assistant Commissioner for Higher and Professional Education 


JAMES SULLIVAN M.A., Ph.D. 


Assistant Commissioner for Secondary Education 


GeEorGE M. Witey M.A., Pd.D., LL.D. 


Assistant Commissioner for Elementary Education 
J.-Cayce Morrison M.A., Ph.D. 


Director of State Library 


James I. Wer M.L.S., Pd.D. 


Director of Science and State Museum 
CHARLES C.. ADAMS M.S’, Ph.D: Disa 
Directors of Divisions 

Administration, LLoyp L. CHENEY B.A. | 
Archives and History, ALEXANDER C. Frick M.A., Litt. D., Ph.D. 
Attendance, 
Examinations and Inspections, AVERY W. SKINNER B.A., Pd.D. 
Finance, CLARK W. HALLIDAY 
Law, IRwin Esmonp Ph.B., LL.B. 
Library Extension, ASA Wynkoop M.A., M.L.S. 
School Buildings and Grounds, FRANK H. Woop M.A. 
Visual Instruction, ALFRED W. AprAms Ph.B. 
Vocational and Extension Education, Lewis A. Witson D.Sc. 


FOREWORD 


Cooperative research holds the key to much of the progress we 
expect during the next decade. The educational measurements move- 
ment has not only given us a new method of working, a new technic, 
but better than this, it means a new attitude. Under its influence 
dogmatism, self-assurance and complacency give way to an inquir- 
ing mind, to an assurance that few questions are ever finally settled 
and to a willingness and desire continuously to seek new truth. 

For the past quarter of a century the spirit of scientific inquiry has 
been gaining an ever deeper hold on the minds of leaders in public | 
school education. The research workers in public school systems, 
and more especially in the colleges and universities, have developed 
a technic of work and methods of procedure. The lifting of the 
entire mass of public school instruction from the level of mere 
opinion to a scientifically evaluated procedure depends, however, not 
so much upon giving teachers and supervisors established and verified 
facts as it does in developing within them the attitude and spirit of 
the research movement. ‘This latter is cooperative research. 

It means much to find 121 rural superintendents, 26 village super- 
intendents and more than 3500 teachers cooperating with the State 
Supervisor of Mathematics and the Educational Measurements 
Bureau in a statewide investigation of achievement and methods of 
teaching arithmetic that is reported in this bulletin, Arithmetic in the 
Rural and Village Schools of New York State. 

It is, and Bauld be, a satisfaction to the school people of the State 
to know that the achievement of the village and rural school pupils 
was seven months above the norm in computation and four months 
above in the solution of reasoning problems. And to those respon- 
sible for the success of children in the rural schools it will be inter- 
esting to note that in arithmetic achievement the one-room schools 
are well above the norm in all grades. 

In the discussion of “practices in teaching arithmetic” in rural 
and village schools, teachers will find much to arouse their interest 
and to encourage them to re-evaluate their whole method of practice 
in the teaching of arithmetic. 

In the text of the report the authors have raised many questions. 
Some of these, teachers and superintendents will want to consider 
in their conference discussions; other questions raised call for addi- 
tional research before adequate answers can be given. They point 
the way toward further cooperative endeavor in the search for truth 
concerning the content and method of arithmetic instruction in the 
elementary schools of the State. 

J. Cayce Morrison 
Assistant Commissioner for Elementary Education 





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University of the State of New York Bulletin 


Entered as second-class matter August 2, 1913, at the Post Office at Albany, 
N. Y., under the act of August 24, 1912. Acceptance for mailing at 
special rate of postage provided for in section 1103, act of 
October 3, 1917, authorized July 19, 1918 


Published Fortnightly 








No. 865 AEDANY. N.Y. December 1, 1926 








« 
ARITHMETIC IN THE RURAL AND VILLAGE 
SCHOOLS OF NEW YORK STATE 


During the month of April 1926 a statewide survey in arithmetic 
was conducted in the villages and rural schools of New York State 
by the Educational Measurements Bureau and the Supervisor of 
Mathematics of the State Department of Education. As in previous 
years the school subject surveyed was chosen after consultation with 
the district superintendents of the State. 

This survey is a timely one since the latest arithmetic syllabus of 
the State Department of Education appeared at the beginning of the 
school year. The survey therefore provides data for comparison of 
achievement in arithmetic with the syllabus requirements and 
standards. 


TNE sty “Wit eet owed BRAS By 


Three phases of arithmetic are included in the survey: computa- 
tion, problem solving, and problem analysis. The Stanford Achieve- 
ment Test, Arithmetic Examination, Form B', was used to measure 
achievement in both computation and reasoning. All pupils included 
in the survey took this test. To measure the ability to analyze arith- 
metical problems the Stevenson Problem Analysis Test? was used. 
This latter test was taken, however, by the pupils in only a few 
schools. 

The test in computation contains 47 examples varying in difficulty 
Mei 3 Perlite. 6 LOSI 129 coc Mang ort? Ueto 
The reasoning (problem solving) test contains 40 problems varying 
in difficulty from “ How many are 5 birds and 4 birds?”’ to “ How 
many cubic feet are there in a cylindrical smoke stack that is 20 feet 
in diameter and 100 feet high?” 

This variation in the difficulty of the material is necessary in order 
to use the same test for all the grades included in the survey. Some 
items must be so easy that the second grade pupils will be able to do 








1 Published by the World Book Co., Yonkers, N. Y. 
2 Published by the Public School Publishing Co., Bloomington, III. 


6 THE UNIVERSITY OF THE STATE OF NEW YORK 


them correctly ; and some of the items must be so difficult that hardly 
any eighth grade pupils will obtain a perfect score. The use of the 
same test for the several grades is advisable in order to be able to 
determine what grade level each pupil or class has attained, and not 
merely whether each pupil or class can obtain a passing mark in a 
test covering a particular grade. 

For each of the tests in computation and reasoning 20 minutes time 
is allowed. The same time is given for all pupils taking the test 
regardless of grade. This makes it possible to compare the achieve- 
ment of pupils in different grades. If two pupils in different grades 
have the same score on the test they are equal in arithmetic achieve- 
ment as measured by this test. If, however, they did not have the 
same time limit such a comparison could not be made. 

The Stevenson Problem Analysis Test consists of six problems. 
With each problem four questions are given referring to the four 
phases of problem solving, namely: what facts are given, what is 
to be found, the process to be used, and the approximate answer. 
Four choices are given for each question and the pupil is to indicate 
in each instance the correct choice. A different form is used for 
grades 7 and 8 than is used for grades 4, 5 and 6. This makes it 
somewhat difficult to compare achievement in problem analysis above 
the sixth grade with achievement in the sixth grade or below. In 
this test the pupils are given as much time as they need in order to 
finish the test. 


ADMINISTRATION OF THE SURVEY 


Twenty-six villages, including 73 schools, and 121 supervisory 
districts in 47 counties participated in this survey. The number of 
villages is approximately the same as in preceding surveys. The 
number of supervisory districts is more than 25 per cent greater 
than any preceding survey and includes almost 60 per cent of all the 
districts in the State. In all approximately 75,000 pupils took the 
tests. This number is more than 35 per cent greater than in any of 
the preceding surveys conducted by the Educational Measurements 
Bureau. 

The tests were given and scored by the teachers, principals or 
superintendents and the class record sheets were sent to the Educa- 
tional Measurements Bureau for statistical treatment. In a number 
of instances the test was given in the second grade. In other 
instances eighth grade pupils, particularly those who passed the 
Regents examination in arithmetic, were not given the test. In 
general, however, where the test was used it was given to pupils in 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 


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8 THE UNIVERSITY OF THE STATE OF NEW YORK 


grades 3 through 7. For purposes of comparison the test records 
are tabulated under five types of schools' — village annual promotion, 
village semiannual promotion, four or more teacher rural schools, 
two or three-teacher rural schools and one-teacher rural schools. 

The four or more teacher classification includes a rather large 
number of schools that are of about the same size as are many of 
the village schools. A separate tabulation was made for these 
schools and the same statistical treatment was given that was made 
for the above-mentioned five types. Since these larger rural schools 
form part of the four or more teacher group they are not treated as 
a separate type in the main body of this report. A supplement to 
this report, p. 22, is devoted to a discussion of the outstanding 
points concerning this type of school. 

Table 1 shows the number of pupils in each grade of each type of 
school taking the Stanford Achievement Test in Arithmetic, com- 
putation and reasoning, and the Stevenson Problem Analysis Test. 


STATISTICAL TREATMENT OF KRESUIS 


The scores on the tests make it possible to answer a number of 
questions, including the following: What is the average grade 
achievement in computation and reasoning in each type of school in 
New York State compared with the standards given for the test? 
How do the ditferent types of schools compare with each other? 
What variation exists between schools of any one type? Does 
achievement in computation compare more favorably with the test 
norms than does achievement in reasoning? How capable are the 
pupils of New York State in analyzing arithmetical problems? How 
does their ability in this compare with their ability to solve problems ? 
To what extent is there overlapping between grades in achievement in 
arithmetic? How do the separate districts and villages compare with 
the norms? 

In order to answer these questions, the test scores were tabulated 
for each village school, or rural district by grades and type of school. 
The distributions thus obtained for the same grade and type of 
school were totalled for the entire State. Medians were computed 
for these total distributions as well as for the separate villages and 
districts. It was also necessary to compute measures of overlapping 
and to tabulate distributions of the median scores for each grade for 
all the districts and villages. 


1 Hereafter these will be referred to as village annual, village semiannual. 
four-teacher, two or three-teacher, and one-teacher schools. 





ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 9 


Table 2 shows the median scores in computation by grades for the 
different types of schools for the entire State. Table 3 shows the 
corresponding data for the reasoning test and table 4 for the test in 
problem analysis. 

The total possible score in computation is 188 points, in reasoning 
160 points, and in problem analysis 24 points. The last line of each 
table gives the norm or standard for each grade for the seventh month 
of the school year, which is about the average time of the giving of 
the tests. 


TABLE 2 


Median scores in computation by grades and types of schools 


TYPE OF SCHOOL ; 3 4 rate 6 P 92 

Annual ...-...seee eee 47.4 72.9 86.4 106.8 124.5 134.4 141.7 

Village ... 4 Semiannual A1......... aresete 7352: aol OeelOS Ot) t246.5 e21315 540014859 
perianal B* 8 as cs vigetee 6404) 8470 995S- 114760 123.8. 14355 

Hour or more teacher.:.. 44.2. 72.7 ° 88.1 106.5 122.3 132.0 142.0 

Rural .... 4 Two or three-teacher.... a>, e0ecta.o. 89 25.510/7.0" 120.4 13220 —14023 
pee COACHET “2. Fea ated se 47.0 970.2, 86.9°° 1036 117.9) 127.21 138.7 

5 


ee Be. ee 76 S60 fit rao sot 


1A =upper half of the school year; B =lower half of the school year. 

2It should be kept in mind in interpreting these data that many eighth grade pupils who 
had passed the Regents examination in arithmetic did not take this test and that therefore 
the eighth grade medians are probably somewhat low. 


TABLE 3 


Median scores in reasoning by grades and types of schools 


TYPE OF SCHOOL P 4 7 Stes ie 6 7 g2 
PRCA So as en giv: «0 40s Bed 2503 5. AS I2- 5° 64 Aa 77 Bre 92.0 2101,9 
Willaver... 4 semiannual Al .....s00.6 See COFOL 51 Oe Ooo OU St 86 ee lLOS cS 
SRL ATNMALS TS? oe cele o a.5'2 peice teOue) 83.00 S57.) 209 ok. 163.4, 101,8 
Four or more teacher... 19.9 "320946584" 62.5—9 80.0. -9027, 101.5 
pete two or three-teacher.... 20.9 31.0 44.6 62.2 76.5 89.7 96.4 
PRO POACHEL 5 oc ae osu a> ewe aero 84s 1o. B O19.) 825.1. 86.2.0 90.9 


DG e Fe Fe sine in 08 SF sie weno 8 13 30 42 56 70 84 $555 


1A = upper half of the school year; B = lower half of the school year, 

2Tt should be kept in mind in interpreting these data that many eighth grade pupils who 
had passed the Regents examination in arithmetic did not take this test and that therefore 
the eighth grade medians are probably somewhat low. 


10 THE UNIVERSITY OF THE STATE OF NEW YORK 


TABLE 4 


Median scores in problem analysis by grades and types of schools 





GRADES 


TYPE OF SCHOOL 4 5 6 7 8 
Four ‘ox more. teachers ove. os eve cas Lae tae ese 11.1 Rey 19.5 20.0 20.0 
Two..or: three-teacher. 5-55; 5% eevee. One eae es 10.7 12.6 18.1 17.6 ee | 
One-teacher)), Vn. ta cies ae bare hb oe ae eee anes bl i3cs 17.0 16.7 18.1 


Norma i868 kk eee ee ae eS ee 16.6 19.4 18.3. 20.2 


The data shown in these tables are presented graphically in figures 
1,2 and 3. It should be noted that the Problem Analysis Test was 
not given at all in the village schools and that there are no data given 
for the second grade for the village semiannual promotion schools. 
Because of the small number of cases the records for the second 
grade in the annual and semiannual promotion schools were combined. 

It is obvious from the data, particularly from the graphs, that 
the differences between the types of schools in any one grade are 
slight in computation and reasoning. The types of schools differ 
more from each other in the upper grades than in the lower. On 
the whole, the medians are highest for the village semiannual pro- 
motion schools and lowest for the one-teacher schools, though this 
is not uniformly the case. The computation medians are in every 
instance higher than the grade norms. The reasoning medians are 
higher than the norms in all but two instances. On the whole, the 
computation medians are much more above the norms than are the 
reasoning medians. All the medians for problem analysis, except 
for grades 6 and 7 of the four or more teacher schools, are below 
the grade norms. The four or more teacher schools are appreciably 
superior to the two or three-teacher and the one-teacher schools and 
are practically at the norm. The latter two types are about equal 
and average about nine months below the norms. 


Table 5 shows the average extent to which each type of school and | 


all the schools together exceed or fall below the norms. The New 
York State schools are, on the average, from five months to nine 
months above the nationwide standards in computation, and from one 
month to seven months above in reasoning. For all types of schools 
taken together arithmetic achievement in the villages and rural schools 
of New York State is almost eight months above the nationwide 
standard in computation, more than four months above in reasoning, 
and in the rural schools almost six months below in problem analysis, 
as measured by tests used in this survey. 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 


Village dem ‘Annual 

Vi lage Annual 

Aor more teacher 
—-— 2-3 Ceacher 


ame = Lleacher 





I iti Ww Vv VI WW WL 


Figure 1—Norms and Median Scores in Computation for 
the Different Types of Schools, Stanford Achievement Test 
Arithmetic Examination, April 1926 


11 


12 THE UNIVERSITY OF THE STATE OF NEW YORK 


| 10 


100 


90 


80} 


AO 
Vell © Semi-A y 
30 Ville Ana ae 
nee 4 ormore leacher 
—-+— 2-Sleacher 
4 —— fteacher 





TL NE YL VIL VIL 
Carades 


Figure 2—Norms and Median Scores in Reasoning for the Different 
Types of Schools, Stanford Achievement Test, Arithmetic Examination, 
April 1926 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 


Aor more leacher 
—.— 2-3 leacher 
— — {leacher 

oOrms 


Ww VI Wile e yuk 





Figure 3— Norms and Median Scores on the Stevenson Problem 
Analysis Test for the Different Types of Schools (Test I for 
Grades 4, 5 and 6; Test II for Grades 7 and 8), April 1926 


13 


14 THE UNIVERSITY OF THE STATE OF NEW YORK 


TABLE 5 


Average grade difference between medians for the different types of 
schools and the norms, and average grade difference for all types 
together in computation, reasoning and problem analysis. Differences 
are shown as months of school achievement 





TYPE OF SCHOOL 


VILLAGE RURAL 
TEST ao H(A SJ XO _s AVERAGE 
Semi- Four- Two orthree- One- 
Annual annual teacher teacher teacher 
Computations wieritske ete 8.6 9.044 8.1 iso 5.5 Fd. 
Reasonine. 1. wie vel set ote ee 5.6 6.6+4 eal 2.9 alee 4.3 


Problemyanaly sist... se Le ie ee ues See aS —8.5 —9I.5 —5.9 


1The norms for the test are given only up to grade 10.0. In some instances median 
scores were higher than this norm and therefore could not be stated exactly in terms of 
grade standing. The differences based on them are therefore given with a + sign in this 
and following tables. 


It is significant that the excess over the norm is almost twice as 
great for computation as it is for reasoning and that in comparison 
with these problem analysis shows up very poorly. Ability in com- 
putation is more mechanical, while ability in problem solving, and 
even more so in problem analysis, involves intelligence more in the 
form of reasoning. Probably the poor standing in problem analysis 
is an explanation of the extent to which problem solving achievement 
in the State falls below achievement in computation. 

The extent to which the medians for the State as a whole exceed 
the norms in both computation and reasoning raises the question as 
to whether on the whole the stress on arithmetic in the State is not 
too great in comparison with other subjects, and whether too much 
time is being spent on the mechanics of arithmetic in comparison with 
the time spent on reasoning. Data from previous surveys indicate 
that, at least in the past, achievement in other school subjects in New 
York State does not compare favorably with achievement in arith- 
metic as shown by the present survey. It would be wise for super- 
intendents and principals in those localities where achievement in 
arithmetic is high to investigate achievement in other subjects in order 
to determine whether stress on arithmetic is at the expense of other 
school subjects. 


It can be seen from table 2 that the grade median for computation 
for the lower half of the grade in the village semiannual promotion 
schools is in each grade higher than the norm for the end of the 
grade. In reasoning (table 3) the medians for the lower half of 
the grade in the village semiannual promotion schools are in no 
instance appreciably below the norm for the end of the grade. 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 15 


Table 6 shows the difference in computation between the several 
types of schools in terms of months of the school year. The table 
shows the extent to which the village annual medians exceed or fall 
below the medians for the other types of schools. The table reads 
as follows: In the second grade the village annual median exceeded 
the four-teacher median by one month, the two or three-teacher 
median by one month, and was the same as the one-teacher median. 
In the third grade the village annual median was the same as for the 
village semiannual, the four-teacher, and the two or three-teacher, 
but was one month higher than the one-teacher median. In the 
fourth grade the village annual median was three months lower than 
the village semiannual median, 1.5 months lower than the four- 
teacher median, etc. 

Table 7 gives the corresponding data concerning the medians for 
reasoning, and table 8 the data for problem analysis. 


TABLE 6 


Differences by grades between types of schools in computation, showing 
the number of months of school achievement by which the village 
annual medians exceed or fall below the medians for the other types 





of schools 

THE VILLAGE ANNUAL IN GRADE 
PROMOTION MEDIANS EXCEED THE Z 3 4 5 6 7 8 AVERAGE 
Village semiannual medians by... 0 —3 —1 0 8 —? —? 
Four or more teacher medians by.. il Q- > —T1.5 Oe reas 5 3 0 ate 
Two or three-teacher medians by.. 1 0 —2.5 O45 3 3 Hod 
One-teacher medians by.......... 0 1 —1 Pp eC 8 5 3.1 

TABLE 7 


Differences by grades between types of schools in reasoning, showing 
the number of months of school achievement by which the village 
annual medians exceed or fall below the medians for the other types 








of schools 

THE VILLAGE ANNUAL IN GRADE 
PROMOTION MEDIANS EXCEED THE 2 3 a 5 6 ih 8 AVERAGE 
Village semiannual medians by. .... —2 —5 —l 0 3. —3+ —1+ 
Four or more teacher medians by 1 1 —l1 1.5 —1l 1 1 8 
Two or three-teacher medians by 1 3 0 2 1 LA ees, 2:7 


One-teacher medians by........ 1 7 Gye! 3 6250 9 4.3 


16 THE UNIVERSITY OF THE STATE OF NEW YORK 


TABLE 8 


Differences by grades between four or more teacher, and one-teacher 
rural schools in problem analysis, showing the number of months of 
school achievement by which the four-teacher school medians exceed 
the medians for the two or three-teacher, and one-teacher schools 


i 


THE FOUR-TEACHER SCHOOL IN GRADES 


4 5 6 7 8 AVERAGE 
Two or three-teacher medians by..... hi 8 6 12 14 8.2 


One-teacher medians by........ecseee 4 7 10 15 11 9.2 


These tables show the slight difference between the types of schools 
in computation and reasoning and the noticeable differences between 
them in problem analysis. The last column represents an average 
difference between the types of schools. On the whole, the differ- 
ences between both types of village schools and the larger rural 
schools is negligible. The one-teacher schools are lower than the 
village and four or more teacher schools by three months in computa- 
tion and more than four months in reasoning. In problem analysis 
the four or more teacher schools are almost a year above the two or 
three-teacher schools and the one-teacher schools. 

Up. to this point a general picture of the conditions for the State 
as a whole has been portrayed. The fact that the median status for 
the State is higher than the norms does not necessarily mean, however, 
that all the villages or districts are above the norms or that all pupils 
are placed in the grades consonant with their arithmetic ability. 
Table 9 gives a measure of the extent to which pupils are not properly 
graded in arithmetic. In order to simplify the presentation only the 
data for the village annual promotion schools are given. The figures 
for the other types of schools are very much the same. ‘The table 
reads as follows: Of the second grade pupils who took the test 3.0 
per cent had scores higher than the third grade computation median; 
6.1 per cent had scores higher than the third grade reasoning median; 
O per cent had scores higher than the fourth grade computation 
median; and .5 per cent had higher scores than the fourth grade 
reasoning median. Of the third grade pupils who took the test 14.8 
per cent had higher scores than the fourth grade computation median; 
21.8 per cent than the fourth grade reasoning median; .3 per cent 
than the fifth grade computation median; and 2.4 per cent than the 
fifth grade reasoning median; 13.8 per cent had scores lower than 
the computation median of the second grade; and 18.0 per cent lower 
than the reasoning median of the second grade, etc. 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 17 


TABLE 9 


Overlapping between grades in village annual promotion schools in com- 
putation and reasoning, showing the per cent of pupils in each grade 
whose scores exceed the median of the next and second higher grades 
and the per cent whose scores are below the medians of the next and 
second lower grades 





PER CENT EXCEEDING PER CENT BELOW 
MEDIAN OF MEDIAN OF 
GRADE TEST (a 
Next higher Second higher Next lower Second lower 
grade grade grade grade 

ba (computation. ...... 350 OnOe 1? Meira, ac te aePtaniescuste 
IGaASONING.. os 6.66 0 6.1 Eom Pieters ol Mei ba Ae: 5 
3 Computation. ........ 14.8 0.3 13 S8e5 weeks acceler. 
REASONING {2 062). o'. 21.8 2.4 LS; Oubet + 2 Sebsgclotones 
4 Computation... .... 21.9 0.4 6.3 322 
WReasOnInNG.... 2.56% 24.5 3.4 5 a oe 6.3 
5 Computation....... Toast =f eiaal 14.9 3.4 
REASONING: 5.4.0... 18.5 2.9 ARS ES 
6 Computation...... ; 22.0 14.6 2One 3.6 
Reasoning: ... 26 so: 28.8 rene! 18.8 6.0 
7 Womoutation. ~..... Sf i PRCA se as Boe 9.7 
PRECISE foc. crs sie ss ZOSIV AY weet ete 28.0 10.7 
8 Mtn ICAeI Otc te wh ec ciclec ale = kaise poe 40.8 302 
SENT er NS ba Sues aN whee 34.1 TSO 


The overlapping in arithmetic seems to be great. In comparison 
with figures for other school subjects, however, it appears that the 
overlapping between grades in arithmetic achievement is probably the 
least of all school subjects. In comparison, for example, with English 
achievement the overlapping in arithmetic is on the whole less than 
one-fourth as great.? 

It is evident from table 9 that the overlapping increases with the 
grades and is greater for reasoning than it is for computation in the 
lower grades but that the opposite is true for the upper grades. 
The overlapping over two grades is very much less than over one 
grade particularly in the lower grades. Undoubtedly these conditions 
are due to the emphasis placed on arithmetic, particularly computa- 
tion, as a basis for grade classification and promotion especially in the 
lower grades, and to the absence of strict uniformity as to the grade 
in which the Regents examination is given. Hence in many instances 
doubling up starts early in many of the upper grades. 

The one outstanding fact that appears from this table is the lower- 
ing in the per cents of overlapping in the neighborhood of the fourth 
and fifth grade in both reasoning and computation. This condition 
is common to all the types of schools. It probably indicates a change 





1See table 6, p. 19, University of the State of New York Bulletin, 846, 
English in the Rural and Village Schools of New York State. 


18 THE UNIVERSITY OF THE STATE OF NEW YORK 


in policy with regard to increased emphasis on arithmetic achieve- 
ment, presumably in the sixth grade. It may also be explained by 
the large per cent of retarded pupils usually found in the fifth and 
sixth grades and by the fact that there is considerable natural over- 
lapping through grades 1 through 4 where the four fundamental 
operations upon integers are taught and reviewed in each grade while 
in grades 5 and 6 distinctly new and more difficult topics are taken up. 


TABLE 10 


Showing for each grade the range in years of school achievement nec- 
essary to include the middle 50 per cent of the scores in each grade 
in computation and reasoning 








TYPE OF SCHOOL TEST 2 3 4 6 7 8 
Village annual .......+. } Gomputaton Fe oe 8 ee 
Village semiannual ..... ) Rompuratyon:- is 41d 16 ee or 
Four or more teacher... ] Romemon; 1g 1s 18 18 La) 2L0b 
Two or three-teacher.....f Romputsvons  -f 18 a7 Te is 23) Blab 
One-teacher +++. s+seeees 1 erations gu }eke Gg ag) 0g ae 
Average seesseeseeeeess | eens ee ie 4:8 8 | zie 


The variation between pupils in achievement in the same grade, 
_ as shown by table 10, is greater in the higher than the lower grades. 
On the whole, the variation is greater in reasoning than in computa- 
tion in the lower grades. The opposite is true in the upper grades. 
The differences in variation between types of schools are negligible. 
The increase in variability with increase in grade is probably due to 
the greater emphasis on arithmetic in the lower grades, particularly 
on computation, as a basis for promotion and classification. 


The extent to which the individual districts and the village schools 
compare with the norms and with each other is of greater importance 
locally than the comparison between the state medians and the norms. 
The distributions of the medians for the separate localities show that 
in computation 13.1 per cent of 586 grade medians for one-teacher 
schools are below the grade norms whereas 26.8 per cent of the 586 
grade medians for one-teacher schools are above the norms for the 
next higher grade. On the other hand, in reasoning 211 of the 587 
grade medians for one-teacher schools are below the grade norms 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 19 


TABLE 11 


Comparison of local medians with grade norms 





MEDIANS ABOVE 
MEDIANS BELOW THE NORM FOR THE 
TYPE OF SCHOOL TEST TOTAL NORMS FOR GRADE NEXT HIGHER GRADE 
NUMBER OF 
GRADE MEDIANS Number Percent Number Per cent 


Village annual ...... eee 189 Spothas ants bab Welaees 
Village semiannual .. | SomP--- 71 Palle Eye See 
Four or more teacher. nae pete = on ee pe 
Two or three-teacher.. | 50mP--- 3a ploy as Sy ee ote 
One-teacher -+...... SR Resiges 389 aiearicine’s en pore 











and 27 are above the norms for the next higher grade. The data 
for the other types of schools are also given in table 11. There- 
fore, although it may be stated that the status of the State as a 
whole in arithmetic is very good, it is still true that in many 
localities it is poor and in a number it may be characterized as too 
good: that is, it is probable that some schools are stressing arithmetic 
disproportionately. 

The variation between localities in arithmetic achievement in the 
same grade can be shown by the range in years of school achievement 
necessary to include the middle 50 per cent of the medians for the 
grade. 


ARLE AZ 


Range in years of school achievement necessary to include the middle 50 
per cent of the median scores in each grade 





GRADE 





TYPE OF SCHOOL? TYPE 2 3 4 5 6 7 8 

Village annual .......... § Computation? 2.). .8 Bd ey LS 2s. A EB 9 
{ Reasoning... é.... wine 1.0 4 ip PS Gale wih 

Foor orimore-teacher.... ' Computation.... .4 .6 Ws ts Moye, ahs 6+ 
Reasoning...... oO 375 oe 2 .6 oe LEO 

Two or three-teacher..... § Computation.... .4 .6 ais .6 5052 Oe pleat 
l Reasoning...... mae Biel .4 0 FY Ae ee EEK 4 

Preteaeher te § Computation.... oa .6 .4 EL Pee nae be ee he es 
| Reasoning...... 12 G5 a: 2 A 245-120 








+The data for the village semiannual schools are omitted because the small number of 
medians for each half year (all less than 20) would make the results too unreliable. 











The differences between schools are on the whole much less in the 
lower than in the upper grades and much less in the rural than in the 
village schools. The amount of variation is in many instances sur- 
prisingly great, particularly in the upper grades. For instance, in 


20 THE UNIVERSITY OF THE STATE OF NEW YORK 


the seventh grade in two or three-teacher schools, the computation 
median for the district ‘that is 25 per cent below the highest is two 
years higher than the median for the district that is 25 per cent above 
the lowest. Of course, the difference between the highest and lowest 
medians is still greater, in this instance being over four and a half 
years. The highest seventh grade computation median among the 
two or three-teacher schools (for a district) is above the tenth grade 
norm. The lowest is below the sixth grade norm. 

Such variation must be recognized to appreciate the fact that many 
localities are doing poorly in arithmetic and many are probably spend- 
ing too much time on arithmetic in comparison with the time spent 
on other subjects, although from tables 2 and 3 it would seem at a 
first glance that achievement for the State as a whole in arithmetic is 
quite satisfactory. 


SUMMARY 


1 Arithmetic achievement in New York State for the entire State 
is high in comparison with the standards for the country as a whole, 
as measured by the Stanford Achievement Test, Arithmetic Examina- 
tion, in both computation and reasoning. 


2 On the average the schools of the State are seven months above 
the norms in computation and four months above in reasoning. 


3 The differences between the grade medians and the norms are 
greater in the lower than in the upper grades. 


4 The medians for the lower half of each grade in the village 
semiannual promotion schools are above the norms for the end of the 
grade in every instance in computation and either at least equal to or 
barely below the norms for the end of the year in reasoning. 


5 The differences between types of schools in median achievement 
in arithmetic are not large. On the whole, the village semi- 
annual promotion schools are highest. The village annual promotion 
and the four-teacher schools are only a little behind. The largest 
difference is between the village semiannual promotion schools and 
the one-teacher schools — an average difference of more than three 
months in computation and more than five months in reasoning. 


6 The one-teacher schools are almost uniformly lowest, although, 
as indicated, the differences are slight. 


7 In problem analysis, a test taken by rural schools only, the 
achievement is low particularly for the two or three-teacher and the 
one-teacher schools, which are on the average about a year below 
the norms. 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS Zi 


8 The differences between the types of rural schools are more 
marked in problem analysis than in either computation or problem 
solving. | 

9 There is a great deal of overlapping between grades in arith- 
metic, the higher the grades the greater the overlapping. 

10 The overlapping is greater for reasoning in the lower grades 
and for computation in the upper grades. 

11 The overlapping in arithmetic is much less than it was found 
to be in English and is probably less than in any other school subject. 

12 The variation between scores in each grade is great, the range 
necessary to include the middle 50 per cent of pupils being from one 
year to more than two and a half years above the second grade and 
only slightly less than a year in the second grade: that is, before 
pupils have had two years of schooling there is a variation in arith- 
metic ability found to be almost a year for the middle 50 per cent of 
the pupils. | 

13 The variation in pupil achievement is greater in the upper than 
in the lower grades. 

i4 The variation in pupil achievement is greater for reasoning in 
the lower, and for computation in the upper grades. 

15 There is no appreciable difference between types of schools 
in the variation in pupil achievement in either computation or 
reasoning. 

16 Although the median achievement for the State as a whole is 
high in both computation and reasoning, 12 per cent of the district 
and village grade medians are below the norms in computation and 
25 per cent are below the norms in reasoning. 

17 On the other hand, 34 per cent of the district and village grade 
medians are above the norm of the next higher grade in computation 
and 13 per cent are above the norm of the next higher grade in 
reasoning. 

18 The variation between grade medians for any one type of school 
is considerable, as shown by the range including the middle 50 per 
cent. The range above the fifth grade varies from six months to 
over two and a half years. Below the fifth grade the variation is 
from two months to 1.2 years. 

19 The variation between districts is least for the one-teacher 
schools and appreciably more for the two or three-teacher and the 
four or more teacher schools. The variation for the villages is very 
much greater on the whole. This may indicate the extent to which 
the types of schools tend to deviate from the syllabus. 


22 THE UNIVERSITY OF THE STATE OF NEW YORK 


RESULTS FOR THE LARGEST RURAL SCHOOLS 


The group of rural schools referred to throughout the foregoing 
as four or more teacher schools includes a number of schools that are 
under district superintendents but that are about as large as many 
of the schools in villages having superintendents. It is possible that 
these should be treated separately from the rest of the four or more 
teacher group although this has not been done in previous surveys. 

A separate tabulation of those rural schools having 15% or more 
teachers shows some rather interesting facts. The accompanying 
tables show the median scores for the village annual promotion 
schools, for the largest? of the four or more teacher schools and for 
all the four or more teacher schools, together with the norms. Table 
13 gives the data for computation and table 14 for reasoning. These 
tables are comparable to tables 2 and 3. 


TABLETS 


Median scores in computation for certain types of schools 


GRADES 


TYPE OF SCHOOL 2 3 4 5 6 he 8 
Village annual promotion........ 47.4 72.9 86.4: 106.8% 124. 5°eeig4seeeeeee 
All four or more teacher........ 44.2. 7 2a7ee Z88et 106.5 1223 132.0 142.0 


Largest of four or more teacher?. a5 74.7 86.9 104.3 12154 13h 141.9 
UNO LTS @ accha isc slo te ae eee aaa 24 57 76 96 114 123 132.5 


1 See footnote 2 at the bottom of this page. 





TaBLe 14 
Median scores in reasoning for certain types of schools 
GRADES 
TYPE OF SCHOOL 2 3 4 : 6 7 8 
Village annual promotion........ PV Be ys OM CC: 64.4 Ar eh: 92.0 2 AGTSY 
All four or more teacher........ 19.9. - 32.9 46.8 62.5 80.0 90.7 | 101.5 


Largest of four or more teacher?. L957 OO tee AO 61.5 79.9 89 S82 eab02eT 
NOYMIS Voom tees. cao Set ee 13 30 42 56 70 84 95.5 





1 See footnote 2 at the bottom of this page. 


In the second and third grades in computation and the third and 
eighth grades in reasoning the medians for the largest rural schools 
are slightly higher than the medians for all the four or more teacher 


1A few schools having less than 15 teachers were included for special 
reasons. 

2 The group. of schools termed “largest of the four or more teacher schools ” 
or “largest rural schools” refers to those schools, under district superintendents, 
that have 15 or more teachers. 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 23 


schools. In every other instance the median for the largest rural 
schools is below the median for all the four or more teacher schools. 
In no instance, however, is the difference so great as two months of 
school achievement. The largest schools were included in obtaining 
the medians for all the four or more teacher schools. The differences 
then between the largest rural schools and the rest of the four or 
more teacher group would be almost twice as great. These differ- 
ences would still, however, be almost negligible. | 

The differences between the medians for the largest rural schools 
and for the village annual promotion schools are also very small, the 
average difference being less than one month of school achievement. 

Table 15 shows the variability of the same three types of schools. 
This table is similar to table 12. 


ABLE LS 


Range in years of school achievement necessary to include the middle 50 
per cent of the median scores in each grade 








GRADE 





TYPE OF SCHOOL 2 3 4 5 6 7 8 

Computation 

Village annual promotion..... Saar 8 ns 12 1.4 1.8 9 

All four or more teacher..... .4 .6 ny, Aff 6 1.6 6+ 

Largest four or more teacher?. .4 ais .6 We 8 TO 1 
Reasoning 

Village annual promotion..... egsae a eRe, .4 9 1.1 fo AY 4 

All four or more teacher..... 3 75 5 5 .6 won A 0 

Largest four or more teacher!?. is 1.0 a 6 6 1.5 1.4 


1 See footnote 2 on page 22. 


The variation between localities in the two types of rural schools 
is almost the same except in grades 7 and 8 in computation and 
grades 3, 7 and 8 in reasoning. On the whole there is a slight 
tendency toward greater variation between localities having larger 
schools, this tendency becoming appreciable in the upper grades. 
In other respects the interpretation for this table is the same as that 
given for table 12. 

One might conclude from the above that there is no appreciable 
difference on the whole between the largest schools under district 
superintendents and those rural schools having from 4 to 15 teachers. 
This is the fact in arithmetic. It must be kept in mind, however, 
that the differences between all types of schools appeared to be very 
small in arithmetic. This, however, might not be true in other 
school subjects. 


24 THE UNIVERSITY OF THE STATE OF NEW YORK 


PRACTICES IN TEACHING ARITHMETIC IN RURAL 
AND VILLAGE SCHOOLS | 


There is much information regarding the teaching of arithmetic in 
the schools of New York State which can not be gained from study- 
ing the results of standardized tests given in this subject. This 
information, however, is extremely valuable in that it offers possible 
suggestions not only for improving the teaching of arithmetic but 
indirectly for improving the teaching of high school mathematics. 
For example, standardized tests will not disclose any uniformity or 
lack of uniformity in the teaching of certain of the fundamental 
operations, in the time devoted to the study of arithmetic or in the 
use of the state syllabus in arithmetic; they will disclose little knowl- 
edge of classroom technic or of the method of handling many prob- 
lems that come up daily in the teaching. 


Sources of Data 


In order to obtain some of this additional information regarding 
the teaching of arithmetic a questionnaire on the prevailing practice 
in teaching arithmetic in the public schools of New York State was 
sent out to teachers in one-teacher, two or three-teacher, and four or 
more teacher schools.t. The following pages contain an attempt to 
summarize the answers to this questionnaire and to make certain 
deductions and suggestions regarding the teaching of arithmetic in 
this State. 

In many instances it was evidently from the answers given that 
some of the questions were misinterpreted or that the teacher filling 
out the questionnaire desired to tell a “‘ good story” and did not put 
down the facts. All such answers were eliminated in the following 
summary. This fact combined with the fact that more than forty 
counties and 162 schools are represented in answers to the question- 
naires make the findings reasonably reliable and representative. 

Table 16 shows the number of replies received from the different 
types of schools. In making the tabulations all of these were used. 


TABLE 16 


Number of replies to the questionnaire from the various schools 


TYPE OF SCHOOL NUMBER OF QUESTIONNAIRES RECEIVED 
One-teacher 1. 4.5605 «232 Ce RE ee. LG 1S. pee 87 
Two or three-teacher.... ..iie wash accede hc tus cn lode eee 16 
Four or more teacher...... \ small village 2.2... oo. . 2s Janes 42 


Ularge village’ .... 2. b). 17 








1A copy of the questionnaire appears on pages 34-36 at the end of this 
report. 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 25 


TABLE 16a 


Number of classes in each grade and the total registration per grade for 
all the schools represented by the answers to the questionnaire 


— 





GRADE NO. OF CLASSES == TOTAL REGISTRATION — 

3 74 343 
4 84 657 
5 103 767 
6 97 886 
7 94 891 
8. 74 607 
} Teaching Load 


Table 17 shows the minimum, maximum and average enrolments 
in each of the grades from 3 to 8 for the various types of schools. 
It was frequently found that the maximum or minimum enrolments 
(more frequently the maximum enrolment) differed quite materially 
from the next nearest enrolment. The average, however, is fairly 
representative of the enrolments for the various grades. 








TABLE 17 
Enrolment by grades in the various schools 

GRADE 3 GRADE 4 GRADE 5 
TYPE OF SCHOOL Min. Max. Av. Min. Max. Av. Min. Max. Av. 
One-teacher viii soo ee 1 16 3 1 9 3 1 12 3 
wo or three-teacher....... 7 16 12 4 15 9 2, ie HH 
Four or more teacher...... 8 45 22 6 43 26 5 43 26 

GRADE 6 GRADE 7 GRADE 8 
TYPE OF SCHOOL Min. Max. Av. Min. Max. Av. Min. Max. Av. 
ROS Tener © asics cco ose 1 8 3 1 7 2 1 5 2 
Two or three-teacher....... 5 12 8 1 12 6 1 12 4 
Four or more teacher...... 5 46 29 4 48 26 9 39 24 


It is interesting to note that the average enrolment of the one- 
teacher school where the teacher has all grades from 3 to 8 inclusive 
is considerably less than the average enrolment in any grade of the 
four or more teacher school. Furthermore, since only about 50 per 
cent of the one-teacher schools have all grades and about 50 per cent 
of the teachers-in-the- four or more teacher schools have more than 
one grade, it would be safe to say that the number of pupils in a one- 
teacher school is about half the number of pupils under a single 
teacher in a four or more teacher school. This statement may be 
verified from the data in table 18. 


26 THE UNIVERSITY OF THE STATE OF NEW YORK 


TABLE 18 


The number of grades for each teacher in the various schools represented 
in the survey 


NO. OF GRADES TWO OR 


PER TEACHER ONE-TEACHER THREE-TEACHER SMALL VILLAGE LARGE VILLAGE 
5 or more 44 3 1 0 
4 20 8 0 0 
3 BS 4 5 1 
2 9 2 15 3 
1 0 0 24 14 


This table shows that nearly 73 per cent of the teachers in one- 
teacher schools teach four or more grades each while in the four or 
more teacher schools nearly 90 per cent of the teachers have no more 
than one or two grades each. 


Time Distribution 


Table 19 shows the time devoted to arithmetic in minutes a week 
in the one-teacher and village schools. As in table 17 it was 
often the case that the maximum (or minimum) time spent in one 
or more of the grades obtained in but one school and the next 
nearest time allotment differed quite materially. This fact, com- 
bined with the fact that in many schools, especially in the one-teacher 
school, the time allotment was estimated, makes this table less reliable 
than some of the other tables. Nevertheless the range of 35 to 225 
in the one-teacher school and of 75 to 450 in the village (four or 
more teacher) school, though these figures be only approximate, is 
far too great and indicates a lack of anything approaching uniformity 
in the matter of time given to arithmetic in the various grades. The 
median time allotment given in this table perhaps comes nearer the 
prevailing practice than would the average. 














TABLE 19 

Minutes a week devoted to arithmetic in the various schools 

GRADF 3 GRADE 4 GRADE 5 
TYPE OF SCHOOL Min. Max. Med. Min. Max. Med. Min. Max. Med. 
One-teacher RA ects tis 35 225 75 45 225 75 45 225 75 
Villaced.a47 0.8 cc tne aes ie CYAGY ol ans ie 75 450 - 150 75° * 450i ea 50 

, GRADE 6 GRADE 7 GRADE 8 
TYPE OF SCIIOOL Min. Max. Med. Min. Max. Med. Min. Max. Med. 
i a ee 
One-teacher’ Sst ies ce? 50 240 75 60 240 80 SOL sees 100 
Village 2. ee ae 7597450.) 200 75) 2400-2200 150 600 # 200 


nn EE Eee 
ool 


——$—$— “gd 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS Zh 


It is reasonable to suppose that the minimum allotment is fairly 
accurate — at least not too small. This means in some schools only 
nine minutes per day is given to arithmetic in grades 4 and 5, 10 
minutes in grade 6, 12 minutes in grade 7 and 10 minutes in grade 8. 
It is interesting to note that according to the medians grade for grade 
the rural (one-teacher) schools are giving less than one-half of the 
time for arithmetic that the village (four or more teacher) schools 
are giving. This fact helps to explain why pupils entering high 
school from rural schools are generally not as well grounded in arith- 
metic, particularly in analysis, as are pupils coming from the village 
elementary schools. 

Table 20 gives an analysis of how the time, in minutes a week, 
given to arithmetic, as indicated in table 19, is divided in the various 
schools and grades between oral arithmetic, problems and drills. 
Here again some of the data had to be approximated as many teachers 
did not make clear-cut distinctions between these three phases of the 
work and accordingly their daily program does not indicate a definite 
amount of time given to each. The data are sufficiently accurate, 
however, to show general practices in these phases of the work and to 
indicate again the unusual range and lack of uniformity among the 
various schools and grades. 


TABLE 20 
Analysis of time spent in arithmetic in the various schools 
GRADE 3 GRADE 4 GRADE 5 
TYPE OF SCHOOL Min. Max. Med. Min. Max. Med. Min. Max. Med. 

Oral arithmetic 

Onre-teacher oo ciss sco eo 5 60 25 10 60 25 5 90 20 

VOLE RS 3 5a oe PR sepa, Ree Seo 15 85 50 15 100 30 
Problems 

One-tezcher <........- 5 150 20 10 150 20 10 150 30 

MEETS) | 2c otra Sicieds  LkAntd < 0 200 50 30 200 80 
Drills . 

One-teacher .......... 5 70 20 5 110 20 5 50 20 

PMs a oe, ae walle so Bese aha hate Chie 10 150 50 10 90 30 

GRADE 6 GRADE 7 GRADE 8 
TYPE OF SCHOOL Min. Max. Med. Min. Max. Med. Min. Max. Med. 

Oral arithmetic 

(ne-Feacher 55.3.2. 06s 5 60 20 10 60 25 5 60 25 

Wiliam nn ptat. fkts<cck ls 20 200 50 20 75 50 25 fide 50 
Problems 

Oneteacher ..=....... 10 120 390 10 150 a5 10 150 40 

WES: 5 aa ee 30 150 90 45 220 100 75 200 100 
Driils 

@ne-teachér ........+. 5 5 15 5 80 20 5 50 15 

GU etc a 20 OO 50) 20 we 50 25 125 50 


28 THE UNIVERSITY OF THE STATE OF NEW YORK 


Judging from the medians in this table the time spent on problems 
equals or exceeds the time put on drills in the one-teacher schools in 
grades 3 to 6 although these are supposedly the grades in which skill 
in the fundamental operations is acquired and problem work con- 
sidered secondary. 

There is no approach to uniformity in the time spent by pupils in 
school and at home in preparation for the recitation in arithmetic. 
That the teachers in many instances had little or no idea of the time 
spent on arithmetic at home could easily be inferred by the fact 
that this item was omitted, indicated by a question mark or by the 
word “ doubtful.” Other vague replies to this question were: “A 
few pupils are willing to do home work.” “ They seldom take books 
home.” ‘ Not much at home” etc., indicating that in many schools 
home work was optional with the pupils. Thirty-three schools 
definitely stated that no home work was given in arithmetic. In 
the other schools the time spent a week at home varied from 15 
minutes to 300 minutes. The time spent in school varied from 20 
minutes to 480 minutes a week or from 4 minutes to 96 minutes 
each day. 

This same lack of uniformity existed also in the schools in which 
the teacher had only the seventh or the eighth grade but to a slightly 
less degree. Thus in these schools the time spent on arithmetic in 
school ranged from 45 minutes to 300 minutes in grade 7 and from 
40 minutes to 300 minutes in grade 8, while the time spent at home 
ranged from O to 150 in each grade, there being six of these schools 
where it was specifically stated that no home work is given. 


Subject Matter 


Seventy-eight teachers reported that they had a separate book for 
oral arithmetic while 70 teachers reported that they had not. Ninety- 
seven teachers indicated that they had available no book on the teach- 
ing of arithmetic other than their text, while 43 indicated that they 
had available other books on the teaching of arithmetic. This latter 
number was very liberal as it included all instances where any book 
other than the text was mentioned although many of these were out 
of date or only indirectly connected with the teaching of arithmetic.? 

In the main, schools are using the new syllabus in arithmetic in 
whole or at least in part. Those not using it give various reasons 
why: “ Plan to start next year,” “not following the syllabus in 
8-A,” “Prepare for Regents at end of seventh year,” “ Following 


1A bibliography of books on the teaching of arithmetic is given on p. 37. 





ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 29 


old syllabus, have no new one,” etc. Uniformity among schools in 
this particular respect is of course highly desirable. 

Standardized tests in arithmetic had been given in all but five of 
the village schools reporting. In the one-teacher schools, however, 
29 had never given any standardized tests in arithmetic although 
these are the schools where such tests are most needed and helpful. 
Time drills are given in most schools to determine individual diffi- 
culties and in most cases these are followed by specific drill. Much 
time is wasted at this point, however, by treating these special drills 
as class drills and not individual drills. 


Method of Procedure in Classes 


Of the schools reporting 89 insist, in teaching the four funda- 
mental operations, that the work be checked; 31 do not insist upon 
the check. 

In the matter of uniformity of method in teaching the fundamental 
operations in the various grades, 13 teachers stated positively that 
there is no such uniformity. Many others did not seem to know if 
there is or is not. These latter teachers apparently teach their own 
way in their grade irrespective of whether it was taught in the same 
Way in previous grades. Most teachers stated, however, that there is 
uniformity in this respect throughout the grades. 

There seemed to be outstanding difficulties in teaching the four 
fundamental operations and there was rather general agreement as 
to what these difficulties are. - In addition, the outstanding difficulty 
seemed to be in those problems involving carrying either in passing 
from one column to the next or in “ going up”’ a column of figures. 
In subtraction the outstanding difficulty seemed to be in teaching 
“borrowing ” and “ paying back.” Several reported difficulty where 
there is a 0 in the minuend. 

It is apparent from the answers to this question that both the 
“take away’ and the “addition” method of subtraction are exten- 
sively used. In multiplication, most teachers reported that a O in 
the multiplier was the greatest stumbling block. A close second 
difficulty was noted in the “carrying” involved in multiplication. 
Several other teachers indicated that they had much trouble in the 
correct placing of partial products in multiplication. In division 
the outstanding difficulty seemed to be in determining the proper 
quotient figure in the various stages of a long division example. A 
great many teachers considered the matter of remainder in division 
their chief problem while still fewer indicated that a 0 in the quotient 
was their chief trouble. 


30 THE UNIVERSITY OF THE STATE OF NEW YORK 


Teaching Problems 


In teaching problems 148 of the teachers indicated that they 
attempted to make their problems informational and the numbers 
real. Most of the teachers stated that they attempted to suit the 
problem to the community. This they claimed to do by basing 
problems on such topics as are tabulated below: 


Child activity Coal 

Community activity Groceries 

Community industries | Farm 

Shopping and marketing Real estate 
Transportation Automobiles 

Lumbering Gas and electricity 
Dairying Potatoes and hay 
Chicken raising Stock in local corporation 


From this list it should be possible to make projects to fit any 
community whether rural or village. Ten teachers indicated that 
they made no effort to suit the problems to the community. There 
were 106 teachers who stated that they gave specific attention to 
problem analysis, while 25 indicated that they did not. 

There were 75 teachers who stated that they were correlating 
problem analysis and silent reading in English, while 59 indicated 
that they were making no effort to do so. It is interesting to note 
that in the one-teacher schools, where this correlation could be done 
best and most economically, it is least done. 

The practice of estimating answers to problems before the solution 
is started is not general. There were 33 teachers who claimed that 
they did this but 103 teachers stated they did not. Checking answers 
to problems is a more general practice. There were 86 teachers who 
claimed that they insisted upon checks. Even here, however, 38 
teachers claimed they did not call for checks. 

One question under the topic “ Teaching Problems ” called forth 
some interesting and enlightening answers. The question was: 
“In problem analysis what have you done to eliminate such incorrect 
labeling of steps as (a) 5 rds & 4 rds = 20 sq rds; or (b) 16 pks ~ 
4—4bu?” One teacher said she saw nothing wrong in either as 
they are. Another teacher said she started in bravely but gave it up. 
Many teachers did not answer this question, leaving one to infer 
that they had no definite plan of correcting this error. In answer to 
(a) some teachers said they “insist upon using abstract numbers 
until the answer is obtained and then label it”; others use the form 
“area = 4 X 5 sq rds.” These two replies are somewhat similar 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 31 


and are perhaps the best ways to treat this kind of a problem. Other 
incorrect replies to (a) show how hazy many teachers are upon this 
point. Some of these replies follow: 
“Teach them you multiply figures not names.” 
“In setting down problems vertically we name the top number 
what we wish to get in our answer.” 
“5 rods length 4 rds width = 20 sq rds area.” 

The teacher giving the last answer tried to camouflage the error by 
inserting the words “length,” “width” and “area.” In reply to 
(Db) one teacher said she “ considers this statement correct.’’ Another 
teacher said she wished she “ knew how to overcome this latter.” A 
few teachers suggest the form 16 pks — 4 pks = 4, the number of bu. 
Below are given some “ model ” forms suggested by other teachers : 


% of 16 pks = 4 pks or 1 bu. (The answer, however, is 4 bu) 
16 pks + 4 = 4, the number of bushels 
16 pks — 4 = 4 X& 1 bu = 4 bu. 
4 pks = 1 bu, 16 pks ~ 4 pks = 4 bu. 
There is little wonder that incorrect forms still exist in the solution 
of such problems when teachers themselves fall into the same error in 
analysis. 


Home Work 


Of the teachers sending in replies to the questionnaire 21 stated 
definitely that they gave no home work in arithmetic. One teacher 
said that she gave home work on Monday evening only. Another 
teacher answered that she did not “allow pupils to do arithmetic 
at home as they get so much help and don’t understand them when 
they get to school.” This latter teacher, it might be noted, gave no 
time to supervised study. 

The method of treating home work in class is not uniform. There 
were 72 teachers who said that they had all home work put on the 
board and explained each day. In general the home work is put on 
the board and read by the pupil although in 9 cases the teacher her- 
self goes through the solution put on the board. In general, home 
work is handed in each day but there is no agreement as to when in 
the recitation period this is done. There were 66 teachers who have 
home work handed in at the beginning of the recitation while 31 
waited until the end of the recitation. It would seem in general 
that the beginning of the period —or early in the recitation is the 
preferable time. 

There were 77 teachers who stated that it was definitely under- 
stood that the first part of the recitation is devoted to questions on 


32 THE UNIVERSITY OF THE STATE OF NEW YORK 


home work, and 129 teachers who stated that they encouraged pupils 
to come to them for help before or after school. While this pro- 
cedure may be commendable and effective if skilfully carried out, 
yet it can easily be the source of much harm. It is this understand- 
ing on the part of the pupil that the first part of the recitation will 
always be devoted to answering questions on the home work that 
tends to make him extremely dependent and willing to give up at 
the first approach of a difficulty. 


Aims 

In stating what they considered the chief aim in teaching arith- 
metic in grades 1 to 6 and in grades 7 and 8 most teachers were very 
vague, clearly indicating that these aims were not well formulated in 
their own minds. Very few teachers made any effort to correct 
the common arithmetical errors found in algebra and none did it for 
physics. Their principal reason for this was that they were not 
teaching these subjects. Hence the high school teacher and the 
grade teacher never get together to work out a campaign against 
these difficulties. 

Equally vague and illusory were the ways in which these teachers 
of arithmetic aimed to prepare pupils for life or more specifically 
for high school. It is quite evident that most teachers of arithmetic 
need a list of objectives worked out for this subject in which due 
consideration is given to the part this subject with its various topics 
is to play in the future life of the pupil — whether it be in school or 
out of school. 

Recommendations 


1 There should be a definite amount of time and a definite 
schedule given to arithmetic each day. A systematic schedule and a 
systematic use of the time given to arithmetic are more important 
than the amount of time given to the subject. 

2 Provision should be made for work in oral arithmetic and 
problem analysis as well as for mechanical drills even though the 
time given to each is short. Here again it is the systematic, per- 
sistent work that counts. 

3 It is desirable that pupils, especially in the upper grades, be 
required to put a definite amount of time on arithmetic even though 
this amount be small. The teacher should have a definite knowledge 
of the amount of time that pupils need to prepare for the next lesson 
and how it is spent. 

4 Every teacher of arithmetic should have available some good 
up-to-date book on the teaching of arithmetic, a copy of the new 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS o8) 


syllabus in arithmetic and specimen sets of standardized tests for 
both fundamental operations and problem analysis. A good _buibli- 
ography to select from is found in the new syllabus. Before using 
such tests it is especially important for teachers to study the directions 
sheet or manual of directions for the test. 

5 Problem material should be informational, and as far as possible, 
suited to particular communities. Answers to problems should be 
estimated before they are actually obtained and checked afterwards. 
Problem analysis may well be correlated with work in silent reading 
in English. 

6 It is perhaps unwise to set aside a part of each day’s recitation 
for questions on work assigned for the day, and equally unwise to 
encourage pupils to seek help from the teacher unless it is carefully 
impressed and each time proved that this is done as a last resort and 
not a first resort. 

7 It is very desirable that all teachers of arithmetic follow the 
new syllabus especially as outlined for the upper grades, that the 
final examination be taken at the time therein indicated and that the 
outline for 8-A be followed as closely as possible so that pupils will 
enter the first year of high school with a uniform preparation. 

8 It is very desirable that every teacher of arithmetic should have 
in mind definite objectives in the teaching of arithmetic not only for 
the subject as a whole but also for the particular grade or grades 
which she is teaching. 


34 THE UNIVERSITY OF THE STATE OF NEW YORK 


PREVAILING PRACTICE IN TEACHING OF ARITH- 
METIC IN THE PUBLIC SCHOOLS OF NEW YORK 
STATE 


No. of teachers in the school ...... 

Schoolies: Gin nape ae Superintendent |... 1... 0.5 se 
(Make answers brief. In most cases use “Yes” or “No” or 
figures.) 

Grades 
Time allotment 3. 4 - 5 1G 
1 Check the grades of work you are 


teachitie 5.1 ie ae ee eee —- -—- Se OO 
2 Number of pupils in each grade.... — — — —— —— = 
3 Total time devoted to arithmetic 
recitations a week in the grade or 
Srades*you teach? 1,4. st aes oe —- —- —- —- — 
4 How many minutes (approximate if 
necessary) each week in each 
grade is given to 
(a} Oral arithmenccs tse ee aoe eee —- —- —- —- — — 
(b) Problems and problem analysis. — — — — — — 
(c) Drills’Sov Ria ae eee —- —- —- —- — — 
5 How much time is spent by pupils weekly in preparation for the 
recitation in arithmetic? 
(a): in schooli i 28) 2 cee (>) at home’.:.%. See 


Subject matter (Fill in the blanks for the grades you are teaching) : 


1 Have you a separate book for oral arithmetic? .......... 

2 Are pupils ever required to make up problems? .......... 

3 Do you make a conscious selection of material for drills and 
problems to give a maximum of emphasis on the use in later 
life Cc Aa ee 

4 What books on the teaching of arithmetic are available for your 
USEF vo cele die esas s cles suse ede © alesis sleet on 

5 Are you following the course as outlined in the new arithmetic 
Syllabtsifis iron ae dees If not, note any departure 22a 

6 What standardized tests in arithmetic have you given in your 
ClaSSES 2 ois oye’ seve ave bw we ue oie Bare ale talus wes ale 

When were they given? .....5.0..0. 000.00. 
What were your, findings? ........ 9.50.6. os 2 «se 


ARITHMETIC IN RURAL AND VILLAGE SCHOOLS 55 


7 Do you give timed drills periodically? .......... Are they 
PeCMaeRCLIZEU fir ceili st: Do you use results to find indi- 
widual difficulties ?.,..3...% 6. Do you follow them up with 
et Tha a) Coe Are these given to the class or 
thesindiyidual? io... 2... 


Method of procedure tn classes (Fill in the blanks for the grades you 
are teaching) : 
MN TAel  VHCAMENLAIS . oo Lk ks cs ape eee ee ee eee 

(a) Do you insist on all work being checked? .......... 

(b) Do you recognize distinct steps in teaching each of the 
fundamental operations? ........... Do you present 
these steps one ata time? .......... 

(c) What step have you found most difficult to get in 
MM PLOT RMR Seach ctolt poate xls 14.0% wisi o's 0s 
ES a STB IN diel oh 1. Pk oiedaithi's: wa) st )ks d's! 6) 9s le'e eva 
BRAN EEN PN ea, Waele A gt os ws Sig iald wee a ev wee 
OTE Oy gine Sips lh ge en 

(d) Is there uniformity in all grades of the method used in per- 
forming each of the fundamental operations? .......... 

(e) List the three most helpful devices you have used in drills. 

(f) When pupils work at the board in groups are they called 
upon to explain their work or just to get the answer? 

2 Teaching problems 
(a) Do you attempt to make problems informational and the 


EE MCI ENT tet etna en Pe a So F8S vaste a Sin ys Se vO Y vite ot 

(b) Do you attempt to suit the problems to your particular 
SAD a SPELT, po UC OSES ee Aico eae 

(c) Do you drill specifically on problem analysis? ............ 
(d) Do you correlate problem analysis and silent reading in 
Ra lt ee ar Ba CEI Go oS 0d av Se Me eds s2 gine Bae ein, 

(e) Do you insist that all answers be estimated before they are 
Pires LER reaper ; checked after they are found? ....... 

(f) List the three most helpful devices you have used in teaching 
ST re ere es ee i i ey 


(g) In problem analysis what have you done to eliminate such 
incorrect labelings of steps as 5 rds & 4 rds equals 20 


CG) O16) 6 6 6 08 6 2 Oo 8 Ss 4 6 8k OS © soe C6 Owe eaten ee eo ecete eee eeweeeones 


Cr 


36 THE UNIVERSITY OF THE STATE OF NEW YORK 


(h) Do you give drills in solving problems without num- 
pers ct. at ae ; with numbers but without all the 


3 Home work 
(a) Do you give any time to supervised study in arithmetic? .. 
(b) Are all home work examples put.on the board or otherwise 


explained by the pupils each day?’ ... oe) ee Do 
they copy from their papers when putting the work on the 
bOdrd.:icaereee ere In explaining the problems does the 
pupil or teacher read through the solution put on the 
Jaray: ge 8 top oh ae 

(c) Is the home: work handed’ in daily? eee [igor at 
what time in the recitation? .... 2... 0) ee 

(d) Is it understood that the first part of the period is devoted 
to questions on home work? .......... 

(e) Do you encourage pupils to come to you for help either 
before school or after school? ..72. 7233 )3eseee 


Aims 


1 What do you consider the chief aim in teaching arithmetic in 
prades: 1-0 iv.5. viele hae oe ee 1-3. bs «2s 
2 Have you made any effort to determine and correct the common 
arithmetical weaknesses evidenced by pupils in algebra? 
Ar nerdy ; in physics? ..........- If so, how aides 
IS cae eet cee ee tie soe sence isa) c iets te iee an 
3 Have you made any effort to eliminate waste in drills by deter- 
mining specific difficulties through diagnostic tests? ......... 


eooesvpteeoeveeeeeeeeeneeeeeeeeesesveoeos e€ee¢ 6 6 8 0 8 6S 6 6 © Oe) eee ee eee 
esevoeseoeoseeewpeeeetmoaoeseeweeenewme7eee eee ee ee 8 6 8 8 hf 8S 6S ee ee ee eee 


eeoeeeveoeseoeeeoecoesueoeaesvn eee Oseeeveeeeeensmeeageeenges 6 6 48 * 8 Be See ee 


ARID ELMEDVG ENSRURAL AND: VIEPAGE SCHOOLS aN 


PammeGRArHy OF BOOKS ON THE TEACHING OF 
ARITHMETIC 


Brown and Coffman. How to Teach Arithmetic. Row Peterson 

Klapper. The Teaching of Arithmetic. Appleton 

Lennes. The Teaching of Arithmetic. Macmillan 

Lindquist. Modern Arithmetic Methods. Scott Foresman 

McLaughlin and Troxell. Number Projects for Beginners. Lippincott 

McNair. Methods of Teaching Modern Day Arithmetic. Badger 

Overman. Principles and Methods of Teaching Arithmetic. Lyons & 
Carnahan 

Smith, D. E. The Teaching of Arithmetic. Ginn 

Stone. How to Teach Primary Arithmetic. Sanborn 

The Teaching of Arithmetic. Sanborn 

Suzzallo. The Teaching of Primary Arithmetic. Houghton-M/ifflin 

Thorndike. The New Methods in “Arithmetic. Rand McNally 

Psychology of Arithmetic. Macmillan é 








BIBLIOGRAPHY OF STANDARDIZED OBJECTIVE tS rS 
IN ARITHMETIC 


This bibliography may be obtained in mimeographed form from the 
Educational Measurements Bureau of the New York State Department 
of Education. Reference is made to this bibliography on p. 40 of this 
bulletin. 


“4 








3 0112 105878653 


Bulletins and Pamphlets Prepared by the Educational Measurements 
Bureau 

BULLETIN 

NUMBER 

734 Morrison, J. C. Educational Measurements. 1921. Out of print 


764 Morrison, J. C. Spelling in Ra | sie Rural Schools. 1922. Out 
of print 

772 Morrison, J.C. 5 ANN nisistraive Uses Made of Standard Tests 
and Scaleg Wwe ie State ‘ EXPY ork, 1921-22. 1923. Out of print 


784 Morrison, J.C. Tl tandard ‘Tests and Scales in the Platts- 
burg High Sc 1 ee 10 cas {NO 

798 Coxe, W. W. Silent Rev in New York Rural Schools. 1924. 
5 cents yERe' 

806 Morrison, J. a) ornell, W. B. & Coxe, W. W. Survey of the Need 


for Special Schools and Classes in Westchester County, New York. 
1924. 5 cents 


814 Coxe, W. W. & Orleans, J. S. One Year’s Reading Progress in 
New York: Rural Schools. 1925. 10 cents 


819 Coxe, W. W. Organization of Special Classes for Subnormal Chil- 
dren. 1925. 5 cents. Revision of Bulletin 802 


835 Orleans, J. S. Survey of Educational Facilities for Crippled Chil- 
dren in New York State. 1925. 5 cents 


839 Coxe, W. W. & Cornell, E. L. A Study of Pupil Achievement and 
Special Class Needs in Westbury, L. I. 1926. 20 cents 

841 Coxe, W. W. Study of Pupil Classification in the Villages of New 
York State. 1925. 20 cents 


843 Gray, E. A. Manual of Suggestions for the Use of the Phonograph 
in Special Classes. 1926. 10 cents 


846 Orleans, J. S. & Richards, E. B. English in the Rural and Village 
Schools of New York State. 1926. 10 cents 
850 Coxe, W. W. & Richards, E. B. Suggestions for Teaching Silent 
Reading. 1926. 5 cents. Revision of Bulletin 803 
851 Coxe, W. W. & others. Outline of a Course in Educational Measure- 
ments for New York State Normal Schools. 1926. 5 cents 
865 Orleans, J. S. & Seymour, F. E. Arithmetic in the Rural and Village 
Schools of New York State. 10 cents 
Tether, C. H. Hints for Special Class Gardens. 5 cents 
Ability Grouping in Junior and Senior High Schools. Mimeographed. 
Free : 
Suggestions for Reclassifying Pupils in Small Rural Schools. Mime- 
ographed. Free 
Coxe, W. W. Adaptation of Methods and Materials of Instruction 
to Ability Groups. Mimeographed. Free 


Bibliography of Objective Tests in Arithmetic. Mimeographed. Free 
Arithmetic Survey, April 1926. Preliminary Reports and Detailed 
Summary Tables. Mimeographed. Free 





Publications of the State Department of Education are free to schools 
and libraries in New York State. Others may purchase limited quantities 
at the cost of publication listed above. 


The above bulletins and pamphlets should be ordered from and checks 
made pay able to The University of the State of New York, Albany, N. Y. 


